Eilenberg–Niven theorem

It is due to Samuel Eilenberg and Ivan M. Niven.

Let where x, a0, a1, ... , an are non-zero quaternions and φ(x) is a finite sum of monomials similar to the first term but with degree less than n. Then P(x) = 0 has at least one solution.

[1] If permitting multiple monomials with the highest degree, then the theorem does not hold, and P(x) = x + ixi + 1 = 0 is a counterexample with no solutions.

Eilenberg–Niven theorem can also be generalized to octonions: all octonionic polynomials with a unique monomial of higher degree have at least one solution, independent of the order of the parenthesis (the octonions are a non-associative algebra).

[2][3] Different from quaternions, however, the monic and non-monic octonionic polynomials do not have always the same set of zeros.